Roots: Pythagoras of Samos

In The Wizard of Oz, the Scarecrow shows us how intelligent he has become by (mis)quoting Pythagoras’ theorem:

“The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.”

Homer Simpson does a similar thing when he puts on a pair of glasses and tries to convince himself that he is smart.

It would seem that the lasting legacy of Pythagoras of Samos is the formula linking the sides of a right-angled triangle. It could, however, be argued that the actual legacy of Pythagoras is much greater—it’s more than the formula used in contrived situations of ladders being rested against walls or finding the answer to that most fundamental of questions: would the pencil stick out of the top of the pot? His legacy is around us every day…

Our knowledge of Pythagoras’ life is a blend of legend and hearsay. His teachings were of a deliberately oral nature because the Pythagoreans argued that questions could not be posed to books in the same way that they could be asked of a person. By bringing together secondary texts and triangulating the facts, we can be certain that Pythagoras (and the Pythagoreans who followed him) were determined to find order in disorder. Any discoveries attributed to Pythagoras were made as a collective—a hive mentality that the Pythagoreans prided themselves on.

The Pythagoreans believed that numbers were at the heart of everything. They held the ‘quaternary’ in extremely high regard, breaking down their understanding of their surroundings into fours—the Tetractys.

Their laws led to a beautiful and harmonious universe—Cosmos—by adding order to chaos. This is perfectly illustrated using music.

Folklore tells of Pythagoras walking past a blacksmith’s workshop and hearing different tones produced by the smithy hammering a rod of metal. While there may be truth in this tale, we do know for definite that the Pythagoreans experimented with an instrument called a monochord.

This simple instrument comprises of a single string, held taut, with a moveable bridge. When the string is plucked, it produces a note. Versions of this instrument are still used today. One notable advocate of the ‘diddley bow’ is Seasick Steve, who has a song dedicated to it.

Seasick Steve and Pythagoras: a striking similarity?

Investigations by the Pythagoreans took the form of creating order from disorder—they interpreted the multitude of possible positions of the bridge as disorder, but by applying the limiting factor of number to the placement of the bridge, harmony and order could be found. They knew that the shorter the string, the higher the pitch of the note produced. By methodically comparing the sounds produced by various string lengths, they found that the notes created by lengths that could be expressed as ratios using small numbers were the most pleasant and harmonious.

The Pythagoreans took these findings as proof of the interrelation of number, harmony and beauty—that mathematics and music were two sides of the same sheet of paper. They defined their musical notes using the ratios between them, rather than giving them individual ‘names’.

An octave would have the ratio 1:2.

A fifth would have the ratio 2:3, which could be calculated by the harmonic mean of the endpoints of the octave.

A fourth would have the ratio 3:4, which could be calculated by the arithmetic mean of the endpoints of the octave.

All of these intervals were expressed using the first four natural numbers so intrinsic to the Pythagorean school of thought.

By transposing these ratios onto the musical scale that we are familiar with today (known as the diatonic scale), a beautiful symmetry can be seen. Taking an arbitrary string length of 24cm, we can see how dividing the string into intervals produces pleasing harmonies.

It is interesting to note that the geometric mean was ignored—it would produce a number in the ratio that would be most discordant! In our 24cm example, the geometric mean would be $12\sqrt{2}$. The discomfort that Pythagoreans felt with regards to irrational numbers is infamous.

Examples of these intervals can be heard in well-known tunes, such as in the first two notes of each of these melodies:

It’s a shame that the Scarecrow’s intelligence could not be portrayed effectively using Dorothy’s opening song: its link to Pythagoras is certainly more universal than $a^{\hspace{1pt}2}+b^{\,2}=c^{2}$.